Just to pick up on the first point: the “degrees of truth” have various roles for Smith, other than defining truth (I’m not sure he commits himself to the disquotational notion being the *only* thing that has a claim to the title truth—it’s just that it’s there if we need to use it). One primary role is as an expert function for beliefs (or something similar)—basically your credences should match expected degree of truth. And they’re also used in defining the logic.

In the aforementioned paper, I’m giving (the standard) bivalent model theory for propositional , in a second order metalanguage (models are, possibly vague, collections of sentence letters.) The sentences that are true in every such model (determinately) are precisely those provable in . (Interestingly, it is sometimes vague if a sentence is true in every model!)

Other than the determinacy of the T-schema I think there are other reasons to think there should be vague assignments of degrees of truth. For example, consider the metalinguistic Sorites:

|1 is small| = 1

If |n is small| = 1 then |n+1 is small| = 1

|10^100 is small| = 1

I find this Sorites as compelling as the standard one for “small” – and once you’ve admitted vagueness in assignments of degree of truth, it seems we haven’t really succeeded in characterizing vagueness as having “intermediate truth value” after all. If you’re going to allow vagueness in truth value assignments, why not just go for the bivalent semantics which allows vagueness in truth value. (I’m working on a paper on intended model theories for non-classical logics where these arguments are laid out more carefully, if you’re interested.)

A possibly confusing thing about that example was the fact that it might not be truth functional in another language – I was thinking that is also truth functional, but not, like negation, a logical constant.

“A standard approach would be to relativize to worlds, or something similar. “Box(p)” is true iff Necessarily, p —that might be ok, but as it stands it looks like a bit of a strange hybrid between T-theory and model theoretic approaches.”

I was thinking that you didn’t need to be a T-theorist to use a (partially) homophonic metalanguage (another difference is, I assume, that the T-theorist only gives an account of truth, not validity.) One of the problems with using a non-modal metalanguage is that you won’t have an intended model of quantified systems unless you quantified over non actual objects. I was thinking of the T-schema as necessary, (and truth as semantic value 1 – to rule out Smith) which means I shouldn’t get those counter examples. Of course, there are ways of reading “’S’ is true in English” that aren’t rigid (I was thinking along the lines of Kaplan, in “Words”.)

“What I’d like to say is that even if you treat “Necessarily” as a logical constant, the approach I sketch will rightly say that “and” is truth-functional, and “necessarily” isn’t.”

This rules out assignments that give “or” wacky interpretations. But what about the argument that you should rule *in* assignments that are vague or contingent in their assignment of truth values. I wasn’t so much concerned with defending the truth functionality of necessity. But the argument that there are more truth functional connectives in the case of non-classical logics seems compelling to me (if you can entertain this kind of non-classical logician.) At what point do you reject the parallel argument that necessity is truth functional. (So the non-classical logician will say agreement is: f(p)=T iff g(p)=T, where “iff” is the non-classical iff, not definable from disjunction and negation. It’s obvious that we should reject the analogous move where “iff” is interpreted as “necessarily p iff q”. But I’m worried the explanation might go along the lines of “iff” thus interpreted isn’t truth functional.)

In the context of a degree theory, some people want to introduce a disquotational truth predicate—with T(‘p’) always taking the same value as p. (e.g. Nick JJ Smith does this at some point in his recent book, I think). Then whenever p is a middling degree of truth, true(‘p’) is middling degree of truth—meeting your requirement. Again, this doesn’t require vague assignments of degrees of truth.

(The non-standard supervaluationist—yes, I was thinking about people who are keen on “penumbral truth” as the right analysis of truth, and so keep the T-schema. McGee/McLaughlin are one such. The view Williamson attacks at the end of the supervaluational chapter is in this vein.)

I should think some more about the issue you raise above. One thing: in what setting do you want to formulate the puzzle? You’re using “necessarily” in the metalanguage to characterize box in the statement above. But that’s not terribly standard. And it’s not obviously right either—e.g. “water is h2O” might satisfy the LHS but not the RHS (I’m thinking of ||=T as “mapped by the intended interpretation to the true” where “the intended interpretation” is non-rigid). A standard approach would be to relativize to worlds, or something similar. “Box(p)” is true iff Necessarily, p —that might be ok, but as it stands it looks like a bit of a strange hybrid between T-theory and model theoretic approaches.

What I’d like to say is that even if you treat “Necessarily” as a logical constant, the approach I sketch will rightly say that “and” is truth-functional, and “necessarily” isn’t. But to understand what it means to take “necessarily” to be a logical constant, we’d need some particular semantic approach in mind, I reckon. John MacFarlane has some interesting stuff in his dissertation on these issues (within a possible-worlds framework).

I guess one project here is just to understand the truth-functional/non-truth-functional distinction as it applies to logical connectives (construed broadly to include e.g. modal operators). But it sounds like you want something more than that: to have something that classifies all English expressions in the right way. I need to think more about that. Have you looked at all on the literature on opaque vs. transparent contexts to see what sort of criteria Quine, Kaplan et al came up with? It seems a similar kind of issue.

That’s very interesting – I hadn’t thought about letting it be vague what the designated value is in non-supervaluationist contexts. I’ll reply here again when I’ve had time to think about it some more. (BTW, non-standard supervaluationism is the kind that keeps the T-schema right?)

Just quickly – I’m still a bit worried about your proposal for defining truth functionality. “It’s expressible in English that p” is a truth functional in English, but presumably it’s not a logical constant. (It seemed like your definition was relying on the coincidence that most of the truth functional connectives happened to be logical too.)

I need to think some more about the vague truth tables. But I do think that it’s not obvious that we need vague truth-tables, even given the disquotational view you describe.

You say: “I beleive that it’s vague whether p, just in case it’s vague whether “p” is true (this follows from the assertion that the T-schema is determinate.) Thus if there’s vagueness in our langauge, we had better admit assignments such that it can be vague whether f(p)=T.”

I don’t see we need to admit assignments such that it is vague what truth-values they assign. An alternative is to say that all truth-value assignments are precise, and when it’s vague whether “p” is true, it’s vague which truth-value value assignment is the designated one.

In effect, we can resist the quantifier shift from: “it’s vague whether, on the intended interpretation f, f(p)=T” to “on the intended interpretation, it’s vague whether f(p)=T”. The wide-scope reading of the definite description “the intended interpretation” would force interpretations with vague truth values. The narrow scope reading allows vagueness in the description “the intended interpretation” to take the strain.

This is the sort of thing I’d expect “non-standard supervaluationists” to say.

So I’d be tempted to characterize truth-functionality in terms of whether every admissible assingment that agrees with the truth values assigned to components agrees with the truth values assigned to compounds. “Admissibility” is (non-circularly) cashed out in terms of preserving the semantic value of logical constants (of course, identifying the logical constants is a major task). And “agreement” can be understood in the obvious way, given that (arguably) we only need to work with fully precise truth-value assignments, and just say it’s vague which is the designated/intended one.

I’m not sure what a non-classical logician who works with a vague metalanguage in describing truth-values or truth-tables should do. There’s some discussion of the difficulties she faces in Williamson’s Vagueness book, IIRC.